CS246: Mining Massive Datasets Jure Leskovec, Stanford University

Transcription

1 CS246: Mining Massive Datasets Jure Leskovec, Stanford University

2 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 2 Supermarket shelf management Market-basket model: Goal: Identify items that are bought together by sufficiently many customers Approach: Process the sales data collected with barcode scanners to find dependencies among items A classic rule: If someone buys diaper and milk, then he/she is likely to buy beer Don t be surprised if you find six-packs next to diapers!

3 A large set of items e.g., things sold in a supermarket A large set of baskets Each basket is a small subset of items e.g., the things one customer buys on one day TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk A general many-to-many mapping (association) between two kinds of things But we ask about connections among items, not baskets 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 3

4 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 4 Given a set of baskets Want to discover association rules People who bought {x,y,z} tend to buy {v,w} Amazon! 2 step approach: 1) Find frequent itemsets 2) Generate association rules Input: TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk Output: Rules Discovered: {Milk} --> {Coke} {Diaper, Milk} --> {Beer}

5 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 5 Items = products; Baskets = sets of products someone bought in one trip to the store Real market baskets: Chain stores keep TBs of data about what customers buy together Tells how typical customers navigate stores, lets them position tempting items Suggests tie-in tricks, e.g., run sale on diapers and raise the price of beer Need the rule to occur frequently, or no $$ s Amazon s people who bought X also bought Y

6 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 6 Baskets = sentences; Items = documents containing those sentences Items that appear together too often could represent plagiarism Notice items do not have to be in baskets Baskets = patients; Items = drugs & side-effects Has been used to detect combinations of drugs that result in particular side-effects But requires extension: Absence of an item needs to be observed as well as presence

7 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 8 First: Define Frequent itemsets Association rules: Confidence, Support, Interestingness Then: Algorithms for finding frequent itemsets Finding frequent pairs A-Priori algorithm PCY algorithm + 2 refinements

8 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 9 Simplest question: Find sets of items that appear together frequently in baskets Support for itemset I: Number of baskets containing all items in I Often expressed as a fraction of the total number of baskets Given a support threshold s, then sets of items that appear in at least s baskets are called frequent itemsets TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk Support of {Beer, Bread} = 2

9 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 10 Items = {milk, coke, pepsi, beer, juice} Support threshold = 3 baskets B 1 = {m, c, b} B 2 = {m, p, j} B 3 = {m, b} B 4 = {c, j} B 5 = {m, p, b} B 6 = {m, c, b, j} B 7 = {c, b, j} B 8 = {b, c} Frequent itemsets: {m}, {c}, {b}, {j}, {m,b}, {b,c}, {c,j}.

10 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 11 Association Rules: If-then rules about the contents of baskets {i 1, i 2,,i k } j means: if a basket contains all of i 1,,i k then it is likely to contain j In practice there are many rules, want to find significant/interesting ones! Confidence of this association rule is the probability of j given I = {i 1,,i k } support( I j) conf( I j) = support( I)

11 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 12 Not all high-confidence rules are interesting The rule X milk may have high confidence for many itemsets X, because milk is just purchased very often (independent of X) and the confidence will be high Interest of an association rule I j: difference between its confidence and the fraction of baskets that contain j Interest( I j) = conf( I j) Pr[ Interesting rules are those with high positive or negative interest values (usually above 0.5) j]

12 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 13 B 1 = {m, c, b} B 2 = {m, p, j} B 3 = {m, b} B 4 = {c, j} B 5 = {m, p, b} B 6 = {m, c, b, j} B 7 = {c, b, j} B 8 = {b, c} Association rule: {m, b} c Confidence = 2/4 = 0.5 Interest = 0.5 5/8 = 1/8 Item c appears in 5/8 of the baskets Rule is not very interesting!

13 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 14 Problem: Find all association rules with support s and confidence c Note: Support of an association rule is the support of the set of items on the left side Hard part: Finding the frequent itemsets! If {i 1, i 2,, i k } j has high support and confidence, then both {i 1, i 2,, i k } and {i 1, i 2,,i k, j} will be frequent conf( I j) = support( I j) support( I)

14 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 15 Step 1: Find all frequent itemsets I (we will explain this next) Step 2: Rule generation For every subset A of I, generate a rule A I \ A Since I is frequent, A is also frequent Variant 1: Single pass to compute the rule confidence confidence(a,b C,D) = support(a,b,c,d) / support(a,b) Variant 2: Observation: If A,B,C D is below confidence, so is A,B C,D Can generate bigger rules from smaller ones! Output the rules above the confidence threshold

15 B 1 = {m, c, b} B 2 = {m, p, j} B 3 = {m, c, b, n} B 4 = {c, j} B 5 = {m, p, b} B 6 = {m, c, b, j} B 7 = {c, b, j} B 8 = {b, c} Support threshold s = 3, confidence c = ) Frequent itemsets: {b,m} {b,c} {c,m} {c,j} {m,c,b} 2) Generate rules: b m: c=4/6 b c: c=5/6 b,c m: c=3/5 m b: c=4/5 b,m c: c=3/4 b c,m: c=3/6 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 16

16 1/9/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 17 To reduce the number of rules we can post-process them and only output: Maximal frequent itemsets: No immediate superset is frequent or Gives more pruning Closed itemsets: No immediate superset has the same count (> 0) Stores not only frequent information, but exact counts

17 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 18 Count Maximal (s=3) Closed A 4 No No B 5 No Yes C 3 No No AB 4 Yes Yes AC 2 No No BC 3 Yes Yes ABC 2 No Yes Frequent, but superset BC also frequent. Frequent, and its only superset, ABC, not freq. Superset BC has same count. Its only superset, ABC, has smaller count.

18 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 19 We are releasing HW1 today It is due in 2 weeks The homework is long Requires proving theorems as well as coding Please start early Recitation session: Review of probability and proof techniques Tomorrow 4:15-5:30pm in Gates B03

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20 Back to finding frequent itemsets Typically, data is kept in flat files rather than in a database system: Stored on disk Stored basket-by-basket Baskets are small but we have many baskets and many items Expand baskets into pairs, triples, etc. as you read baskets Use k nested loops to generate all sets of size k Note: We want to find frequent itemsets. To find them, we have to count them. To count them, we have to generate them. 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 21 Item Item Item Item Item Item Item Item Item Item Item Item Etc. Items are positive integers, and boundaries between baskets are 1.

21 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 22 The true cost of mining disk-resident data is usually the number of disk I/Os In practice, association-rule algorithms read the data in passes all baskets read in turn We measure the cost by the number of passes an algorithm makes over the data

22 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 23 For many frequent-itemset algorithms, main-memory is the critical resource As we read baskets, we need to count something, e.g., occurrences of pairs of items The number of different things we can count is limited by main memory Swapping counts in/out is a disaster (why?)

23 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 24 The hardest problem often turns out to be finding the frequent pairs of items {i 1, i 2 } Why? Freq. pairs are common, freq. triples are rare Why? Probability of being frequent drops exponentially with size; number of sets grows more slowly with size Let s first concentrate on pairs, then extend to larger sets The approach: We always need to generate all the itemsets But we would only like to count (keep track) of those itemsets that in the end turn out to be frequent

24 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 25 Naïve approach to finding frequent pairs Read file once, counting in main memory the occurrences of each pair: From each basket of n items, generate its n(n-1)/2 pairs by two nested loops Fails if (#items) 2 exceeds main memory Remember: #items can be 100K (Wal-Mart) or 10B (Web pages) Suppose 10 5 items, counts are 4-byte integers Number of pairs of items: 10 5 (10 5-1)/2 = 5*10 9 Therefore, 2*10 10 (20 gigabytes) of memory needed

25 Two approaches: Approach 1: Count all pairs using a matrix Approach 2: Keep a table of triples [i, j, c] = the count of the pair of items {i, j} is c. If integers and item ids are 4 bytes, we need approximately 12 bytes for pairs with count > 0 Plus some additional overhead for the hashtable Note: Approach 1 only requires 4 bytes per pair Approach 2 uses 12 bytes per pair (but only for pairs with count > 0) 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 26

26 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, bytes per pair 12 per occurring pair Triangular Matrix Triples

27 Approach 1: Triangular Matrix n = total number items Count pair of items {i, j} only if i<j Keep pair counts in lexicographic order: {1,2}, {1,3},, {1,n}, {2,3}, {2,4},,{2,n}, {3,4}, Pair {i, j} is at position (i 1)(n i/2) + j 1 Total number of pairs n(n 1)/2; total bytes= 2n 2 Triangular Matrix requires 4 bytes per pair Approach 2 uses 12 bytes per occurring pair (but only for pairs with count > 0) Beats Approach 1 if less than 1/3 of possible pairs actually occur 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 28

28 1/9/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 29 Approach 1: Triangular Matrix n = total number items Count pair of items {i, j} only if i<j Keep pair counts in lexicographic order: Problem is if we have too {1,2}, {1,3},, {1,n}, {2,3}, {2,4},,{2,n}, {3,4}, many items so the pairs Pair {i, j} is at position (i 1)(n i/2) + j 1 Total number do not of fit pairs into n(n 1)/2; memory. total bytes= 2n 2 Triangular Matrix requires 4 bytes per pair Can we do better? Approach 2 uses 12 bytes per pair (but only for pairs with count > 0) Beats Approach 1 if less than 1/3 of possible pairs actually occur

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30 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 31 A two-pass approach called A-Priori limits the need for main memory Key idea: monotonicity If a set of items I appears at least s times, so does every subset J of I Contrapositive for pairs: If item i does not appear in s baskets, then no pair including i can appear in s baskets So, how does A-Priori find freq. pairs?

31 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 32 Pass 1: Read baskets and count in main memory the occurrences of each individual item Requires only memory proportional to #items Items that appear ss times are the frequent items Pass 2: Read baskets again and count in main memory only those pairs where both elements are frequent (from Pass 1) Requires memory proportional to square of frequent items only (for counts) Plus a list of the frequent items (so you know what must be counted)

32 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 33 Item counts Frequent items Main memory Counts of pairs of frequent items (candidate pairs) Pass 1 Pass 2

33 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 34 You can use the triangular matrix method with n = number of frequent items May save space compared with storing triples Trick: re-number frequent items 1,2, and keep a table relating new numbers to original item numbers Main memory Item counts Frequent items Old item #s Counts of pairs of Counts frequent of pairs items of frequent items Pass 1 Pass 2

34 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 35 For each k, we construct two sets of k-tuples (sets of size k): C k = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k 1 L k = the set of truly frequent k-tuples All items Count the items All pairs of items from L 1 Count the pairs To be explained C 1 Filter L 1 Construct C 2 Filter L 2 Construct C 3

35 ** Note here we generate new candidates by generating C k from L k-1 and L 1. But that one can be more careful with candidate generation. For example, in C 3 we know {b,m,j} cannot be frequent since {m,j} is not frequent 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 36 Hypothetical steps of the A-Priori algorithm C 1 = { {b} {c} {j} {m} {n} {p} } Count the support of itemsets in C 1 Prune non-frequent: L 1 = { b, c, j, m } Generate C 2 = { {b,c} {b,j} {b,m} {c,j} {c,m} {j,m} } Count the support of itemsets in C 2 Prune non-frequent: L 2 = { {b,m} {b,c} {c,m} {c,j} } Generate C 3 = { {b,c,m} {b,c,j} {b,m,j} {c,m,j} } Count the support of itemsets in C 3 Prune non-frequent: L 3 = { {b,c,m} } **

36 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 37 One pass for each k (itemset size) Needs room in main memory to count each candidate k tuple For typical market-basket data and reasonable support (e.g., 1%), k = 2 requires the most memory Many possible extensions: Association rules with intervals: For example: Men over 65 have 2 cars Association rules when items are in a taxonomy Bread, Butter FruitJam BakedGoods, MilkProduct PreservedGoods Lower the support s as itemset gets bigger

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38 Observation: In pass 1 of A-Priori, most memory is idle We store only individual item counts Can we use the idle memory to reduce memory required in pass 2? Pass 1 of PCY: In addition to item counts, maintain a hash table with as many buckets as fit in memory Keep a count for each bucket into which pairs of items are hashed For each bucket just keep the count, not the actual pairs that hash to the bucket! 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 39

39 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 40 New in PCY FOR (each basket) : FOR (each item in the basket) : add 1 to item s count; FOR (each pair of items) : hash the pair to a bucket; add 1 to the count for that bucket; Few things to note: Pairs of items need to be generated from the input file; they are not present in the file We are not just interested in the presence of a pair, but we need to see whether it is present at least s (support) times

40 1/9/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 41 Observation: If a bucket contains a frequent pair, then the bucket is surely frequent However, even without any frequent pair, a bucket can still be frequent So, we cannot use the hash to eliminate any member (pair) of a frequent bucket But, for a bucket with total count less than s, none of its pairs can be frequent Pairs that hash to this bucket can be eliminated as candidates (even if the pair consists of 2 frequent items) Pass 2: Only count pairs that hash to frequent buckets

41 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 42 Replace the buckets by a bit-vector: 1 means the bucket count exceeded the support s (call it a frequent bucket); 0 means it did not 4-byte integer counts are replaced by bits, so the bit-vector requires 1/32 of memory Also, decide which items are frequent and list them for the second pass

42 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 43 Count all pairs {i, j} that meet the conditions for being a candidate pair: 1. Both i and j are frequent items 2. The pair {i, j} hashes to a bucket whose bit in the bit vector is 1 (i.e., a frequent bucket) Both conditions are necessary for the pair to have a chance of being frequent

43 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 44 Item counts Frequent items Main memory Hash Hash table table for pairs Bitmap Counts of candidate pairs Pass 1 Pass 2

44 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 45 Buckets require a few bytes each: Note: we do not have to count past s #buckets is O(main-memory size) On second pass, a table of (item, item, count) triples is essential (we cannot use triangular matrix approach, why?) Thus, hash table must eliminate approx. 2/3 of the candidate pairs for PCY to beat A-Priori

45 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 46 Limit the number of candidates to be counted Remember: Memory is the bottleneck Still need to generate all the itemsets but we only want to count/keep track of the ones that are frequent Key idea: After Pass 1 of PCY, rehash only those pairs that qualify for Pass 2 of PCY i and j are frequent, and {i, j} hashes to a frequent bucket from Pass 1 On middle pass, fewer pairs contribute to buckets, so fewer false positives Requires 3 passes over the data

46 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 47 Item counts Freq. items Freq. items Main memory First hash First table hash table Bitmap 1 Bitmap 1 Second hash table Bitmap 2 Counts of of candidate pairs pairs Pass 1 Pass 2 Pass 3 Count items Hash pairs {i,j} Hash pairs {i,j} into Hash2 iff: i,j are frequent, {i,j} hashes to freq. bucket in B1 Count pairs {i,j} iff: i,j are frequent, {i,j} hashes to freq. bucket in B1 {i,j} hashes to freq. bucket in B2

47 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 48 Count only those pairs {i, j} that satisfy these candidate pair conditions: 1. Both i and j are frequent items 2. Using the first hash function, the pair hashes to a bucket whose bit in the first bit-vector is 1 3. Using the second hash function, the pair hashes to a bucket whose bit in the second bit-vector is 1

48 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, The two hash functions have to be independent 2. We need to check both hashes on the third pass If not, we would end up counting pairs of frequent items that hashed first to an infrequent bucket but happened to hash second to a frequent bucket

49 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 50 Key idea: Use several independent hash tables on the first pass Risk: Halving the number of buckets doubles the average count We have to be sure most buckets will still not reach count s If so, we can get a benefit like multistage, but in only 2 passes

50 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 51 Item counts Freq. items Main memory First First hash hash table table Second hash Second table hash table Bitmap 1 Bitmap 2 Counts Counts of of candidate candidate pairs pairs Pass 1 Pass 2

51 1/8/2014 Jure Leskovec, Stanford CS246: Mining Massive Datasets, 52 Either multistage or multihash can use more than two hash functions In multistage, there is a point of diminishing returns, since the bit-vectors eventually consume all of main memory For multihash, the bit-vectors occupy exactly what one PCY bitmap does, but too many hash functions makes all counts > s